-
Title: Complex Logarithm
-
Series: Complex Analysis
-
YouTube-Title: Complex Analysis 13 | Complex Logarithm
-
Bright video: https://youtu.be/VBX43M1VeoE
-
Dark video: https://youtu.be/gU-PY-oCuZs
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ca13_sub_eng.srt missing
-
Timestamps
00:00 Intro
00:12 Logarithm as inverse of exponential function
00:33 How it works in the real numbers
01:54 Complex exponential function
03:08 exp(iy) lies on the unit circle
04:00 Definition of π
06:05 Exponential is 2πi-periodic
06:49 Exp is not injective
08:40 Restrict to open strip and slotted plane
09:06 Definition of logarithm (principal value)
09:56 Properties of logarithm
12:22 Credits
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Which claim is not correct?
A1: $$ \exp \left( i \frac{\pi}{2} \right) = i$$
A2: $$ \exp \left( i \pi \right) = -1$$
A3: $$ \exp \left( i \frac{3 \pi}{2} \right) = -i$$
A4: $$ \exp \left( i 2 \pi \right) = -1$$
Q2: What is $\exp(i y)$ for $y \in \mathbb{R}$?
A1: $\cos(y) + \sin(y)$
A2: $\cos(y) + i \sin(y)$
A3: $\cos(y) - \sin(y)$
A4: $\cos(y) - i \sin(y)$
Q3: Let $\log$ be the principal value of the logarithm. What is $\log(i)$?
A1: $0$
A2: $\frac{\pi}{2}$
A3: $i \frac{\pi}{2}$
A4: $i \frac{3 \pi}{2}$
-
Last update: 2024-10
